NOTE: Assuming $p > 5$. The result is not true for $p=2, 3$ or $5$. In all the excitement to show, I missed important point.

For $p=2$ obviously $1$ is not divisible by $2$ - does not hold, and
$p=3$, $11$ is not divisible by $3$ and finally for $p=5$ gcd$(5,10)=2$ and therefore not true as well ($1111$ is not divisible by $5$).